If X > 0 does √x have to be > 0?

Question

In a quadratic, the question was
x - 3√x - 10 = 0 and it gave that x > 0

So if I used √x to solve it then (√x - 5) (√x + 2) = 0
then √x = 5 or √x = -2
So x would be 25 or 4.
I'm just wondering now if √x can equal -2, and if someone could explain.
Thanks,

Answer 1

If a² = x then a = ±√x

So √x is of necessity positive. This is the same as as
√3 = + 1.7320508.......
≠±1.7320508.......

The language of mathematics is exact and cannot have two different meanings, otherwise it would not be necessary to use ± in the quadratic formula:

x = (-b ± √(b² - 4ac))/2a

Therefore √x = -2 is not a valid solution because - √x has to equal -2

√x = 2 is the solution of (and √x = -5 is disallowed)

Addendum

1, ,,, for Joe C: x + 3√x - 10 = 0
ie (√x - 2)(√x + 5) = 0
So √x = +2 or √x = -5 Methinks you jumped to an incorrect conclusion

2. x = y² is not a function (it is a parabola with its axis the line y = 0 and vertex at (0, 0) Domain x ≥ 0 Range all y
However it can be broken up into 2 separate functions namely
y = √x Domain x ≥ 0 Range y ≥ 0
and y = - √x. Domain x ≥ 0 Range y ≤ 0
This, of necessity, means √x ≥ 0

3. In realty, the equation x - 3√x - 10 = 0 has a different story to the equation y² - 3y - 10 = 0. This has 2 solutions y = -2 or 5 whereas the previous has only one solution and that is that √x = 5.

Answer 2

Wal C gave a good explanation but reversed his last sentence.

He said,
√x = 2 is the solution of x + 3√x - 10 = 0 = 0 (and √x = -5 is disallowed)

whereas I think he meant,
√x = 5 is the solution of x - 3√x - 10 = 0 = 0 (and √x = -2 is disallowed)

Addendum to Wal C's addendum:
equation is x - 3√x - 10 = 0 not x + 3√x - 10 = 0. I thought you had the best answer in any case.

Answer 3

√x is defined to be a positive real number (when working in the reals). When working in complex numbers, it is not necessarily a real number. Therefore, with the restricted domain of √ to positive numbers, we have a function. Otherwise, we don't.

How do we see that √x=-2 is NOT a solution to your equation:

Let f(x)=x-3√x-10, and g(x)=x+3√x-10.

Then (fg)(x)=(x-10-3√x)(x-10+3√x)= (x-10)^2-(3√x)^2 = x^2-29x+100 (with our domain restricted to x≥0). This is a polynomial function that we all know and love. Since it is a degree 2 polynomial function, it has AT MOST 2 roots (in reals or complex). It is obvious that f and g both have at least 1 root. If one of them had 2 roots, then there would be three roots of the polynomial fg. This is not possible, therefore, there can be only one solution. In particular, there is NO number x (real or complex) such that √x=-2.


Many people have said that √1=±1 since (±1)^2=1. This is not true when working over the real numbers. -1 is a primitive 2nd root of unity (which means that (-1)^2=1, and that all roots of x^2-1 are a power of -1), but when working with real numbers, √1 is defined to be 1. When working with complex numbers, we get some different solutions, thus when working with complex numbers, √x is NOT a function.

Answer 4

you can't take the square root of a negative real number, unless you are willing to deal with complex numbers, but that's another question.

think of this way
-2x-2=4 or -2squared =4
2x2=4 or 2squared=4

now the square root of 4 is 2 or -2. it is possible
The problem is what is the square root of -4, is it 2 or -2?

The main reason the question asked you to only deal x>0 is that it makes the question easier. Also if x<0 you wouldn't be able to solve this question without using complex numbers. I don't knew complex numbers very well so maybe someone else could help you out before I make it to the library.

Answer 5

if x>0 then √x need not be > 0

But then √x = 5 or √x = -2 from your equation.

So when u say that √x = -2, x = -2 * -2 = 4

So 4 = -2 * -2 or 4 = 2 * 2

-ve * -ve = +ve
Thus here you have all the possible values for
x =4 => √x * √x -- eq 1
where √x = 2 or -2

as √x + 2 = 0 => √x = -2

Thus sating eq 1

Answer 6

√x does not have to be >0. since √ of any number is either a positive or negative value. thus if x is>0 . √x is either positive or negative. e.g √4= -2 or +2
so √x can equal -2. remeber negative numbers are always <0

Answer 7

No, sqrt(X) does not have to be greater than 0, only X has to be greater than 0. sqrt(X) can equal -2 because this implies that X = 4 which is greater than 0. Get it?

Answer 8

Yes, the square root of x can be -2, becasue -2 times -2 is 4 which is positive. Negative numbers do not have real roots.

Answer 9

the x>0 is only a condition, so thats means you should choose 5 since -2 did not meet the condition

Answer 10

yes sqrt(x) or the symbol that you have used is > 0. we say solution of

(x-2)^2 = 5 as

x = 2 + sqrt(5) or 2-sqrt(5) to indicate positive square root being taken.